Exact Solutions for a Third‐Order KdV Equation with Variable Coefficients and Forcing Term

Alvaro H Salas S,Cesar A Gómez S

doi:10.1155/2009/737928

Abstract

The general projective Riccati equation method and the Exp‐function method are used to construct generalized soliton solutions and periodic solutions to special KdV equation with variable coefficients and forcing term.

Highlights

It is well known that in an early phase of the development of the solitons theory, there were already many applications in physics and engineering
Some powerful computational methods such as the tanh method 2, the generalized tanh method 3, 4, the extended tanh method 5–10, the improved tanh-coth method 11–13, the Exp-function method 14–18, the modified Exp-function method, the Cole-Hopf transformation, the projective Riccati equation method PREM 21, 22, the generalized projective Riccati equations method 23–25, the extended hyperbolic function method 26, and many other methods have been developed in this direction
We use this last two methods to obtain soliton and periodic solutions to the following special KdV equation with variable coefficients and forcing term: ut α t uux kα t uxxx F t, 1.3 where F t is an external forcing function varying with time t, k is a constant, and α α t is a function of t, α t / 0

Summary

Introduction

It is well known that in an early phase of the development of the solitons theory, there were already many applications in physics and engineering. Traveling waves as solutions of the KdV equation ut 6uux uxxx 0. The equation ut k1tnuux k2tmuxxx 0, 1.2 where k1, k2 are arbitrary constants, which have applications in physics, has been analyzed in 1 from the point of view of its exact solutions. The search of explicit solutions to Mathematical Problems in Engineering nonlinear partial differential equations NLPDEs using analytic methods is not an easy task. We use this last two methods to obtain soliton and periodic solutions to the following special KdV equation with variable coefficients and forcing term: ut α t uux kα t uxxx F t , 1.3 where F t is an external forcing function varying with time t, k is a constant, and α α t is a function of t, α t / 0. Substituting 1.7 into 1.6 , we obtain htVξαtλVξftVξ kλ3α t V ξ 0, 1.8 where fftFt dt

The Exp-Function Method

General Projective Riccati Equation Method

Periodic and Soliton Solutions

Conclusions